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Brunnian links, claspers and Goussarov–Vassiliev finite type invariants

Published online by Cambridge University Press:  01 May 2007

KAZUO HABIRO*
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. e-mail: habiro@kurims.kyoto-u.ac.jp

Abstract

Goussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.

In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Bar–Natan, D.. On the Vassiliev knot invariants. Topology 34 (1995), 423472.CrossRefGoogle Scholar
[2]Birman, J. S.. New points of view in knot theory. Bull. Amer. Math. Soc. (N.S.) 28 (1993), 253287.CrossRefGoogle Scholar
[3]Birman, J. S. and Lin, X. S.. Knot polynomials and Vassiliev's invariants. Invent. Math. 111 (1993), 225270.Google Scholar
[4]Conant, J.. Vassiliev invariants and embedded gropes, preprint.Google Scholar
[5]Gusarov, M. N.. A new form of the Conway–Jones polynomial of oriented links. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 (1991), Geom. i Topol. 1, 4–9, 161. English translation: Topology of manifolds and varieties. Adv. Soviet Math. 18 (1994), 167–172.Google Scholar
[6]Gusarov, M.. On n-equivalence of knots and invariants of finite degree. Topology of manifolds and varieties. Adv. Soviet Math. 18 (1994), 173192.Google Scholar
[7]Gusarov, M. N.. The n-equivalence of knots and invariants of finite degree. Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 208 (1993), 152173; English translation: J. Math. Sci. 81 (1996), 2549–2561.Google Scholar
[8]Goussarov (Gusarov), M.. Finite type invariants and n-equivalence of 3-manifolds. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 517522.CrossRefGoogle Scholar
[9]Gusarov, M. N.. Variations of knotted graphs. The geometric technique of n-equivalence. Algebra i Analiz 12 (2000), 79125; English translation: St. Petersburg Math. J. 12 (2001), 569–604.Google Scholar
[10]Habiro, K.. Claspers and finite type invariants of links. Geom. Topol. 4 (2000), 183.Google Scholar
[11]Kanenobu, T. and Miyazawa, Y.. The second and third terms of the HOMFLY polynomial of a link. Kobe J. Math. 16 (1999), 147159.Google Scholar
[12]Kontsevich, M.. Vassiliev's knot invariants. I. M. Gel'fand Seminar. Adv. Soviet Math. 16, Part 2 (American Mathematical Society, 1993), 137150.Google Scholar
[13]Milnor, J.. Link groups. Ann. of Math. 59 (1954), 177195.CrossRefGoogle Scholar
[14]Miyazawa, H. A. and Yasuhara, A.. Classification of n-component Brunnian links up to C_n-move. Topology Appl. 153 (2006), 16431650.CrossRefGoogle Scholar
[15]Ohyama, Y.. A new numerical invariant of knots induced from their regular diagrams. Topology Appl. 37 (1990), 249255.CrossRefGoogle Scholar
[16]Przytycki, J. H. and Taniyama, K.. The Kanenobu–Miyazawa conjecture and the Vassiliev-Gusarov skein modules based on mixed crossings. Proc. Amer. Math. Soc. 129 (2001), 27992802.CrossRefGoogle Scholar
[17]Stanford, T. B.. Four observations on n-triviality and Brunnian links. J. Knot Theory Ramifications 9 (2000), 213219.CrossRefGoogle Scholar
[18]Taniyama, K.. On similarity of links. Gakujutsu Kenkyuu, School of Education, Waseda University, Series of Mathematics 41 (1993), 3336.Google Scholar
[19]Vassiliev, V. A.. Cohomology of knot spaces. Theory of singularities and its applications. Adv. Soviet Math. 1 (American Mathematical Society, 1990), 2369.Google Scholar